Three-Level Laser Coupled to Squeezed Vacuum Reservoir

We analyze the quantum properties of the light generated by a three-level laser with an open cavity and coupled to a two-mode squeezed vacuum reservoir via a single-port mirror. The three-level laser consists of three-level atoms available in an open cavity and pumped from the bottom to the top level by means of electron bombardment. Applying the large-time approximation scheme, we have obtained the steady-state solutions of the equations of evolution for the expectation values of the atomic operators and the quantum Langevin equations for the cavity mode operators. Using the resulting steady-state solutions, we have calculated the mean photon number, the variance of the photon number, and the quadrature variance for the two-mode cavity light. We have seen that the light generated by the three-level laser is in a squeezed state and the squeezing occurs in the plus quadrature. It so turns out that the maximum quadrature squeezing of the two-mode cavity light is 45 . 68% for M 0 = 0 . 59 and N 0 = 0 . 27 below the vacuum-state level. In addition, we have shown that the effect of the squeezed parameter is to increase the mean and variance of the photon number.


Introduction
Squeezed states of light have played a crucial role in the development of quantum physics. Squeezing is one of the nonclassical features of light that has been extensively studied by several authors [1,2,3,4,5,6,7,8,9,10,11,12]. In squeezed light the noise in one quadrature is below the vacuum-state level at the expense of enhanced fluctuations in the other quadrature, with the product of the uncertainties in the two quadratures satisfying the uncertainty relation [9,10]. Squeezed light has potential applications in low-noise optical communication and weak signal detection [10]. Squeezed light can be generated by various quantum optical processes such as subharmonic generations [1,2,3,4,5], four-wave mixing [2,3], resonance fluorescence [1,8], second harmonic generation [1,2,3], and three-level laser under certain conditions [1,10]. A three-level laser is a quantum optical system in which light is generated by three-level atoms inside a cavity usually coupled to a vacuum reservoir via a single-port mirror. When a three-level atom in a cascade configuration makes a transition from the top to the bottom level via the intermediate level, two photons are generated. If the two photons have the same frequency, then the three-level atom is called degenerate three-level atom otherwise it is called nondegenerate. The squeezing and statistical properties of the light produced by three-level lasers when the atoms are initially prepared in a coherent superposition of the top and bottom levels or when these levels are coupled by strong coherent light have been studied by several authors [13,14,15,16,17,18,19,20]. These authors have found that these quantum optical systems can generate squeezed light under certain conditions. Moreover, Fesseha [9,10] has studied the squeezing and the statistical properties of the light produced by a three-level laser with the atoms placed in a closed cavity and pumped by electron bombardment. He has shown that the maximum quadrature squeezing of the light generated by the laser, operating below threshold, is found to be 50% below the vacuumstate level. Moreover, he has also found that the quadrature squeezing of the output light is equal to that of the cavity light. On the other hand, this study shows that the local quadrature squeezing is greater than the global quadrature squeezing. He has also found that a large part of the total mean photon number is confined in a relatively small frequency interval. In addition, Fesseha [10] has studied the squeezing and the statistical properties of the light produced by a three-level laser with the atoms placed in a closed cavity and pumped by coherent light. He has shown that the maximum quadrature squeezing is 43% below the vacuum-state level, which is slightly less than the result found with electron bombardment.
In this paper, we seek to analyze the quantum properties of light emitted by three-level atoms available in an open cavity coupled to a squeezed vacuum reservoir via a single port-mirror and pumped from the bottom level to the top level by the means of electron bombardment. Thus taking into account the interaction of the three-level atoms with a resonant cavity light and the damping of the cavity light by a vacuum reservoir, we obtain the photon statistics and the quadrature squeezing of the cavity light. We carry out our calculation by considering the interactions of the cavity mode with the thermal reservoir via a single port-mirror and the three-level atoms with the cavity mode as well as a vacuum reservoir.

Operator dynamics
We consider here the case in which N three-level atoms in a cascade configuration and available in an open cavity. We denote the top, intermediate, and bottom levels of these atoms by |a k , |b k , and |c k , respectively. We prefer to call the light emitted from the top level light mode a and the one emitted from the intermediate level light mode b. We carry out our analysis with light modes a and b having the same or different frequencies. In addition, we assume that light modes a and b to be at resonance with the two transitions |a k → |b k and |b k → |c k , with direct transition between |a k and |c k to be electric-dipole forbidden. The interaction of a three-level atoms with cavity modes a and b can be described at resonance by the Hamiltonian [10] whereσ k a = |b kk a|, are lowering atomic operators,â andb are the annihilation operators for the cavity modes, g is the coupling constant between the atom and the cavity modes. We assume that the cavity modes are coupled to a two-mode squeezed vacuum reservoir via a single-port mirror. The quantum Langevin equations for the operatorsâ andb are given by [9,10] where κ is the cavity damping constant andF a (t) andF b (t) are noise operators associated with squeezed vacuum reservoir and having the following correlation properties: With the aid of Eqs. (1), (4), and (5), one can easily establish that Furthermore, the master equation for a three-level atom interacting with a squeezed vacuum reservoir is given by [10] dρ dt where with r being squeezed parameter and γ is the spontaneous emission decay constant. We can rewrite Eq. (12) as whereη k a = |a kk a|, Using Eq. (1), we can put Eq. (15) in the form Now applying the relation d dt along with Eq. (18), we can easily establish that Three-Level Laser Coupled to Squeezed Vacuum Reservoir We see that Eqs. (20)-(25) are nonlinear differential equations and hence it is not possible to find exact time-dependent solutions of these equations. We intend to over come this problem by applying the large-time approximation [13]. Then using this approximation scheme, we get from Eqs. (10) and (11) the approximately valid relationŝ Evidently, these would turn out to be exact relations at steady state. Now combining Eqs. (28) and (29) with Eqs. (20)-(25), where is the stimulated emission decay constant. We next proceed to find the expectation value of the product involving a noise operator and an atomic operator that appears in Eqs. (30) -(35). To this end, after removing the angular brackets, Eq. (33) can be rewritten as wheref a (t) is the noise operator associated withη a . A formal solution of this equation can be written aŝ Multiplying Eq. (38) on the right byF a (t) and taking the expectation value of the resulting equation, we have Ignoring the noncommutativity of the atomic and noise operators and neglecting the correlation betweenF a (t) andσ k a (t ), assumed to be considerably small [6], one can write the approximately valid relations Now on account of these approximately valid relations along with the fact that a noise operatorF at a certain time should not affect the atomic variable at earlier time, Eq. (39) takes the form Following a similar procedure, one can also check that We also take We note that Eqs. (50) -(54) represent the equation of evolution for the atomic operators in the absence of the pumping process. The pumping process must surely affect the dynamics of η k a and η k c . We seek here to pump the atoms by electron bombardment. If r a represents the rate at which a single atom is pumped from the bottom to the top level, then η k a increases at the rate of r a η k c and η k c decreases at the same rate. In view of this, we rewrite Eqs. (52) and (54) as We next sum Eqs. (50), (51), (53), (55), and (56) over the N three-level atoms, so that Three-Level Laser Coupled to Squeezed Vacuum Reservoir in whichm we easily arrive at Furthermore, applying the definition given by Eq. (2) and setting for any k σ k a = |b a|, we havem a = N |b a|.
Following the same procedure, one can also check thatm Moreover, using the definitionm =m a +m b and taking into account Eqs. (70)-(75), it can be readily established that With the aid of Eq. (68), one can put Eq. (59) in the form Applying the large-time approximation scheme to Eq. (60), we get Thus on taking into account this result, Eq. (81) can be written as The steady-state solution of Eq. (83) is expressible as Using the steady-state solution of Eq. (61) along with Eq. (82), we have On account of Eq. (84), Eq. (85) takes the form For r a = 0, we see that N a = N b = 0 and N c = N . This result holds whether the atoms are initially in the top or bottom level. In the presence of N three-level atoms, we rewrite Eq. (10) as [10] dâ in which λ and β are constants whose values remain to be fixed. Applying Eq. (28), we get and on summing over all atoms, we have where stands for the commutator ofâ andâ † when light mode a is interecting with all the N three-level atoms. On the other hand, applying the large-time approximation to Eq. (87), one can easily find Thus on account of Eqs. (89) and (91), we see that In view of Eqs. (92) and (93), Eq. (87) can be written as Following a similar procedure, one can also readily establish that 194 Three-Level Laser Coupled to Squeezed Vacuum Reservoir Furthermore, in order to include the effect of pumping process, we rewrite Eqs. (57) and (58) as in whichĜ a (t) andĜ b (t) are noise operators with vanishing mean and µ is a parameter whose value remains to be determined. Employing the relation d dt along with Eq. (97), we easily find Now comparison of Eqs. (83) and (101) shows that and We observe that Eq. (103) is equivalent to One can also easily verify that Furthermore, adding Eqs. (57) and (58), we have wherem is given by Eq. (77). Upon casting Eq. (106) into the form one can also easily verify that µ has the value given by Eq. (102) and On the other hand, assuming the atoms to be initial in the bottom level, the expectation value of the solution of Eq. (97) happens to be m a (t) = 0.
Hence the expectation value of the solution of Eq. (94) turns out to be In view of Eqs. (94) and (110), we claim thatâ(t) is a Gaussian variable with zero mean. One can also easily verify that Then on account of Eqs. (96) and (111), we realize thatb(t) is a Gaussian variable with zero mean.
In addition, adding Eqs.((94) and (96), we get whereF andm is given by Eq. (77). One can also easily check that In view of Eqs. (112) and (114), we see thatĉ is a Gaussian variable with zero mean.

Photon statistics
In this section we wish to calculate the mean and variance of the photon number for the two-mode cavity light at steady state. To this end, using the relation along with Eq. (114), we readily find Next we seek to evaluate ĉ † (t)m(t) . Applying the large-time approximation, one gets from Eq. (114) the approximately valid relationĉ Multiplying the adjoint of Eq. (122) on the right bym(t) and taking the expectation value of the resulting expression, we get We now proceed to evaluate F † c (t)m(t) . To this end, a formal solution of Eq. (107) can be written aŝ Multiplying Eq. (124) on the left byF † c (t) and taking the expectation value of the resulting expression, we have Taking into account the fact that a noise operatorF at a certain time should not affect the atomic variable at earlier time and assuming that the cavity mode and atomic mode operators are not correlated, we get  On account of this result, Eq. (123) takes the form We next seek to evaluate F † c (t)ĉ(t) . To this end, a formal solution of Eq. (114) can be written aŝ Multiplying Eq. (128) on the left byF † c (t) and taking the expectation value of the resulting expression, we get In view of Eqs. (117) and (126) along with the fact that a noise operatorF at a certain time should not affect the atomic variable at earlier time, Eq. (129) becomes Now on account of Eqs. (127) and (130) along with their complex conjugates, we can rewrite Eq. (121) as The steady-state solution of this equation is expressible as Following a similar procedure, one can establish that In view of Eqs. (82), (84), and (86), Eqs. (132) and (133) can be rewritten as We note that, unlike the mean photon number, the quadrature squeezing does not depend on the number of atoms. This implies that the quadrature squeezing of the cavity light is independent of the number of photons. The plot in Fig. 3 indicate that the maximum quadrature squeezing is 45.68% below the vacuum-state level and this occurs when the three-level laser is operating below threshold .

Conclusion
In this paper we have studied the statistical and squeezing properties of the light generated by three-level atoms available in an open cavity and pumped to the top level by electron bombardment at constant rate. Applying the large-time approximation scheme, we have obtained the steady-state solutions of the equations of evolution for the expectation values of the atomic operators and the quantum Langevin equations for the cavity mode operators.
Using the resulting steady-state solutions, we have calculated the mean photon number, the variance of the photon number, and the quadrature variance for the two-mode cavity light. We have seen that the light generated by the three-level laser is in a squeezed state and the squeezing occurs in the plus quadrature. It so turns out that the maximum quadrature squeezing of the two-mode cavity light is 45.68% below the vacuum-state level. Our result shows that the maximum quadrature squeezing is less than the one obtained by Fesseha [10]. This is due to the vacuum reservoir noise. In addition, we have shown that the effect of the squeezed parameter is to increase the mean and variance of the photon number.